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3 Answers
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For Spear of Heliod, you have a static ability with a continuous effect that grants +1/+1.

For Undying, you have a triggered ability with a one-shot effect that grants a counter that grants +1/+1.

Notice the difference between what grants the +1/+1. It's the effect granting the +1/+1 in one case, and it's the counter in the other case. This matters because the +1/+1 will only be granted as long as the granter exists.

The +1/+1 from the continuous effect will last as long as the continuous effect exists. In the case of the effects from static abilities, that's as long as the card with the static ability is on the battlefield.

The +1/+1 granted by the counter will last as long as the counter exists. These normally go away when the permanent on which they reside ceases to exist, although there are cards that can manipulate counters (e.g. Clockspinning), and granting a -1/-1 counter will cause both the +1/+1 and the -1/-1 counter to be removed.

Another difference is that counters are represented by physical objects (e.g. glass beads or dice used as counters), but effects aren't represented physically.

+1 for "effects aren't represented physically." So many new players insist on putting a die on a creature whenever such an effect comes into play. It is a pet peeve of mine as it can lead to confusion about the board state.
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Fr33danJul 15 '14 at 15:43

Yes, +1/+1 counters are different from effects that give +1/+1. Counters are physical markers that can be added, removed, or moved around. Effects that give +1/+1 are separate from the creature, and go away when the effect runs out or the permanent causing the effect leaves the battlefield.

A one-shot effect happens all at once, and then finishes. Damage, card draw, life gain, and adding counters are all one-shot effects. The important thing here is that the outcome stays around after the effect ends (well, damage gets cleared at the end of turns, but that's a separate rule). This means that once a creature gets a counter, the counter stays on it until it dies or something else does something to the counter.

A continuous ability, on the other hand, usually has either a limited timespan or is tied to a permanent. Spear of Heliod's first ability is a static ability creating a continuous effect. Giant Growth also creates a continuous effect that lasts until the end of the turn. In those cases, your creatures only get the +1/+1 (or the +3/+3) until Spear of Heliod leaves the battlefield, or until your turn ends in Giant Growth's case.

Yes, they are completely different. A counter is a physical object that you place on your creature card to show that some effect is being applied. In your example, both would have the result of increasing your creature's strength and toughness by 1, but counters have all sorts of things that interact with them.